Spring2005_1

Spring2005_1 - MATH 136 Spring 05 Midterm 1 solutions 1...

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MATH 136 Spring 05 Midterm 1 solutions 1. Consider the system of linear equations: (1 + i ) x 1 + 2 ix 2 = 1 (1 + i ) x 2 + x 3 = 1 2 - 1 2 i x 1 - x 3 = a [1] a) Write the augmented matrix for the system. 1 + i 2 i 0 1 0 1 + i 1 1 2 - 1 2 i 1 0 - 1 a [3] b) Row reduce the matrix to row echelon form using elementary row operations. 1 + i 2 i 0 1 0 1 + i 1 1 2 - 1 2 i 1 0 - 1 a 1 - i 2 r 1 r 1 1 1 + i 0 1 2 - 1 2 i 0 1 + i 1 1 2 - 1 2 i 1 0 - 1 a r 3 - r 1 r 3 1 1 + i 0 1 2 - 1 2 i 0 1 + i 1 1 2 - 1 2 i 0 - 1 - i - 1 a - 1 2 + 1 2 i r 3 + r 2 r 2 1 1 + i 0 1 2 - 1 2 i 0 1 + i 1 1 2 - 1 2 i 0 0 0 a [4] c) Determine the value of a which makes the system consistent. For that value of a , ±nd the general solution of the system. From the matrix in part b) we see that we need a = 0 for the system to be consistent. To ±nd the general solution we row-reduce into reduced row echelon form. 1 1 + i 0 1 2 - 1 2 i 0 1 + i 1 1 2 - 1 2 i 0 0 0 0 r 1 - r 2 r 1 1 0 - 1 0 0 1 + i 1 1 2 - 1 2 i 0 0 0 0 1 - i 2 r 2 r 2 1 0 - 1 0 0 1 1 2 - 1 2 i - 1 2 i 0 0 0 0 So we have x 1 = x 3 x 2 = - 1 + i 2 x 3 - 1 2 i x 3 is free 1
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2 [6] 2. a) Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the matrix 1 0
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Spring2005_1 - MATH 136 Spring 05 Midterm 1 solutions 1...

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